"New Vector skin forces to user to click on the tabs on "mw-article-toolbar-container", while it is sufficient to hover over the tabs when you have the Legacy Vector skin."
Same on bnwiki.

https://github.com/wikimedia/Vector/commit/41d30615a24fdfe7da156d7f4be83b6b61428c17 is the responsible commit. Hover states were apparently intentionally kept in legacy Vector and intentionally removed from the new Vector. In my opinion the hover states should not be removed. Having to click is only a minor annoyance, but an annoyance nonetheless (but that's just my opinion :)

I would also very much like the TW tab of the legacy Twinkle to start working again, because the new version is not ready yet.

This is a separate issue that should be filed as a separate bug. I'm not sure the official version of Twinkle supports new Vector yet, so you might be on your own for the time being... but feel free to raise the issue at https://github.com/wikimedia-gadgets/twinkle/issues so that this is at least tracked.

This is by design.
@alexhollender and @Volker_E provided rationale in T275681#6882448

User styles/scripts can be added to restore this where wanted but I don't think this should be the default.

For anyone looking to restore this behavior, this code in your common.css file works for me:

.vector-menu-checkbox:hover ~ .vector-menu-content, .vector-menu-content:hover {
	opacity: 1;
	visibility: visible;
	height: auto;
}

Might be worth a second look at this ticket. It has come up in offwiki discussion.

Interaction with other dropdown-style menus should be considered, since many editors have Twinkle, etc. installed.

With a hover system, it takes 0 clicks to close the Twinkle menu and open the More menu. With a non-hover system, it takes 2 clicks.

Thanks for pointing this out out @Novem_Linguae — this is actually a bug. The TW menu should automatically close when you open the Tools/More menu (or any other menu, or click anywhere else on the page). @Jdlrobson what would the best approach to fixing this be?

Not automatically closing other menus is somewhat intentional - there may be benefit in opening multiple menus at a time (in fact some gadgets automatically do this for all menus) and closing them may be unexpected, in the same way that some people get annoyed when clicking on a link opens a new window rather than the existing window. I think the click behaviour is correct here.

The hover behaviour is also a bit of an anti-pattern but there are gadgets that restore that behaviour (more information on https://www.mediawiki.org/wiki/Reading/Web/Desktop_Improvements/Features/User_menu#Why_doesn't_the_user_menu_show_on_hover?)

The overlapping of the two menus doesn't look good however, so for that I've opened T328792